An implicit numerical scheme for a class of backward doubly stochastic differential equations

TL;DR

This paper develops an implicit numerical scheme for a class of backward doubly stochastic differential equations without forward diffusion, establishing convergence rates and providing explicit solutions for linear cases using Malliavin calculus.

Contribution

The paper introduces a novel implicit numerical scheme for BDSDEs with convergence analysis and explicit solutions for linear cases, expanding computational methods in stochastic calculus.

Findings

Established $L^p$-H"{o}lder continuity of solutions.

Proposed an implicit numerical scheme with proven convergence rate.

Derived explicit solutions for linear BDSDEs with random coefficients.

Abstract

In this paper, we consider a class of backward doubly stochastic differential equations (BDSDE for short) with general terminal value and general random generator. Those BDSDEs do not involve any forward diffusion processes. By using the techniques of Malliavin calculus, we are able to establish the $L^p$-H\"{o}lder continuity of the solution pair. Then, an implicit numerical scheme for the BDSDE is proposed and the rate of convergence is obtained in the $L^p$-sense. As a by-product, we obtain an explicit representation of the process $Y$ in the solution pair to a linear BDSDE with random coefficients.